Perturbation analysis of the extinction probability of a Markovian binary tree

TL;DR

This paper conducts a perturbation analysis on the extinction probability of supercritical Markovian Binary Trees, providing bounds and error estimates for solutions of the associated quadratic vector equation, validated by numerical tests.

Contribution

It introduces a perturbation bound and an error estimate for the minimal nonnegative solution of the quadratic vector equation in MBTs, enhancing understanding of solution stability.

Findings

Bounds are fairly sharp based on numerical tests.

Perturbation analysis provides insights into solution stability.

Error bounds help assess approximation quality.

Abstract

The extinction probability of the Markovian Binary Tree (MBT) is the minimal nonnegative solution of a Quadratic Vector Equation (QVE). In this paper, we present a perturbation analysis for the extinction probability of a supercritical MBT. We derive a perturbation bound for the minimal nonnegative solution of the QVE, and an error bound is also given, which can be used to measure the quality of an approximation solution. Numerical tests show that these bounds are fairly sharp.